To find the time for one revolution (period) and the radius of the circle of a conical pendulum, we use the given information:

The radius of the circular path is related to the length of the string and the angle $\theta$ by trigonometry.

$$r = L \sin\theta$$

Substitute the given values:

$$r = (1.0 \text{ m}) \sin(30^\circ)$$ $$r = (1.0 \text{ m}) (0.5)$$ $$r = 0.5 \text{ m}$$

Converting to centimeters:

$$r = 0.5 \text{ m} \times \frac{100 \text{ cm}}{1 \text{ m}}$$ $$r = 50 \text{ cm}$$

For a conical pendulum, the period $T$ is given by the formula:

$$T = 2\pi \sqrt{\frac{L \cos\theta}{g}}$$

Substitute the given values:

$$T = 2\pi \sqrt{\frac{(1.0 \text{ m}) \cos(30^\circ)}{9.8 \text{ m/s}^2}}$$

We know that $\cos(30^\circ) = \frac{\sqrt{3}}{2} \approx 0.8660$.

$$T = 2\pi \sqrt{\frac{1.0 \text{ m} \times 0.8660}{9.8 \text{ m/s}^2}}$$ $$T = 2\pi \sqrt{\frac{0.8660}{9.8}}$$ $$T = 2\pi \sqrt{0.088367... \text{ s}^2}$$ $$T = 2\pi (0.297266... \text{ s})$$ $$T \approx 1.8676 \text{ s}$$

The radius of the circle is $\boxed{50 \text{ cm}}$.
The time for one revolution (period) is $\boxed{1.87 \text{ s}}$.